Methodology for mathematical and information modeling of complicated systems is developed. A complicated system is considered as a system with a complex nature of interaction between elements. The difference between complicated systems and “large” systems is shown. Complicated systems are studied as control objects. Methodology for mathematical modeling of complicated systems is based on combining of fuzzy logic and classical mathematics. Such combining makes it possible to exclude the participation of experts in the process of the development of the decision support systems. This allows you to avoid the difficulties associated with expert evaluations in organizing decision-making under uncertainty. Methodology for information modeling of complicated systems is based on the method of object-oriented analysis (Shlaer-Mellor method). Particular attention is paid to recommendations for the selection of the object attributes for objects that model complicated systems in the information models. Every value of every object attribute (except identifiers of the object) varies from zero to one as the value of a fuzzy variable. There is an example of developing an information model based on the proposed methodology. This is information model of the potentially detonative object. Information structure diagram for the complex potentially detonative object is composed for general case. Information structure diagrams for different kinds of the potentially detonative object are built in general terms. The proposed methodology is adequate for modern technological processes. It is used successfully for enlargement and improvement of DSS for explosion-proof of the grain processing enterprises of different types. Original software for real time control of risky situations is created.
Object-oriented analysis (OOA) [
Every “all-in-one” domain or subsystem of the
complicated domain must be analyzed in three
steps [
These steps are separate but integrated parts for OOA.
Information modeling is aimed at identification of objects, which make up a system for (object-oriented) analysis.
Every object (class) corresponds to a set of the
real world things. All instances of the object
(elements of the set):
have the same characteristics (that can be
abstracted as attributes [
There are identifiers among attributes, i.e. attributes which values identify each individual instance of an object uniquely.
There are different ways for representation of
an object. It can be presented either graphically or
in tabular form [
It is rather simple to choose attributes for objects that correspond to such real world things which may be described as “simple systems”. But it is not so easy to choose attributes for objects that describe so-called “complicated systems”. It should be borne in mind that (from the practical point of view) values of the object attributes should be calculated relatively easily, and the attributes themselves should describe the state of the object quite accurately.
The aim of this research is to develop recommendations for the selection of attributes for objects that model complicated systems in information models.
Considering the problem of mathematical and information modeling of complicated systems, one should initially define what a complicated system is.
The system is considered as an ordered set of structurally interconnected and functionally interacting elements.
At first glance, it is natural to consider a system
consisting of large number of elements as a
complicated system (according to the principle:
the greater the number of elements, the more
complicated the system is). But this point of view
is obviously not correct. Indeed, a system can
consist of a large number of similar (identical)
elements interacting with each other on the basis
of well-known and simple principles (laws). Such
systems are usually pretty easily described by
statistical laws. For example, an ideal gas [
An ideal gas is a theoretical gas composed of
many randomly moving point particles that are
not subject to interparticle interactions [
Thus, not every "large" system, i.e. a system consisting of a large number of elements, is a complicated system. Although, naturally, a complicated system can be a " large" system.
A complicated system should be considered as a system with a complex nature of interaction between elements and, as a rule, with dissimilar elements.
A complex nature of interaction between elements suggests two different possibilities: Mathematical equations describing this interaction can not be solved either analytically or numerically (at least, the solution of these equations by known numerical methods cannot be carried out in an acceptable time); The interaction between elements of the system cannot be described at all by the equations of classical mathematics due to the difficulties of formalization.
Systems are studied primarily as control objects. Every system interacts with the external environment (other systems or objects) and are characterized by input and output parameters.
Effective control of systems (control objects) in most cases requires the construction of an adequate mathematical and information models of these systems. This is especially true for control process based on the principle of the compensation of perturbations.
From the standpoint of the control theory, it does not matter what the complexity of the interaction of the system elements consists in (which, in fact, allows to consider the system itself as a complicated system).
For mathematical modeling of a control object, which is a complicated system, it is necessary to apply the theory of probability or fuzzy logic. This study considers the second possibility.
Let us consider complicated system as control object, which is determined by n parameters p1, p2,…, pn. These parameters are controlled parameters of this object. A specific set of these parameters defines the object state at the moment.
To construct a mathematical model of the control object means to write down the parameters p1, p2,…, pn that fully determine the state of the object as functions of the other parameters m1, m2,…, mk that determine (from the point of view of the control problem) the state of the control object environment. Parameters m1, m2,…, mk assumed to be known as functions of time t.
Thus there are n functions from k variables i = i( 1, 2, … , k) ( = 1, … , ), (1) where m1, m2,…, mk are functions of time t.
So n functions pi (i=1,…,n) are composite functions from time t: pi(m1(t),m2(t),…,mk(t))=pi(t)(i=1,…,n). (2) The main problem of this approach is that for complicated systems it is almost impossible to define functions (1) and, as a consequence, to define functions (2). It's almost impossible even if the functions mj(t) (j=1,…,k) are accurately defined. This impossibility is usually connected with a very complicated nature of physical (mechanical, chemical) models of the control object itself (if it is a technical or technological object) and the processes in which this object participates. These processes reflect, among other things, the interaction of the object with the environment.
It should also be noted that the parameters p1, p2,…, pn may not be independent.
As a matter of fact, in some cases it is not possible to ascertain the presence or absence of the corresponding relations between these parameters. In addition, in a number of cases it makes sense to consider the obviously interdependent parameters of an object in order to organize effective control of this object.
Thus there are q (q<n) functions from n variables ri = ri(p1, p2,…, pn) (i=1,…,q), (3) but this fact is not essential.
The proposed methodology of developing of mathematical model for a complicated system includes the following items.
Finding of simplified mathematical
relations between pi (i=1,…,n) and mj
(j=1,…,k), i.e. construction of functions pi =
fi(m1, m2,…, mk) (i=1,…,n). As a result, the
values of the parameters pi are found only
approximately. Solving of this problem is the
most difficult part of the proposed method
realization. It requires deep knowledge of the
technical/technological process, mathematics
and special sciences (mechanics, physics
and/or chemistry).
Finding of intervals [pimin, pimax] for
possible changes of pi (i=1,…,n). pimin is
minimum value of pi for the
technical/technological process as a whole.
Accordingly, pimax is maximum value of pi for
the technological process as a whole. As a rule,
pimin and pimax are determined by production
regulations and technical propeties and
capabilities of equipment.
Replacement of every value pi by the
corresponding interval [pi*, pi**] (i=1,…,n),
where the inequalities pi*< pi**, pi*< pimax and
pimin < pi** take place, but the inequality pimin <
pi*< pi**< pimax is not always correct. Usually
the length of the interval [pi*, pi**] is much less
than the length of the interval [pimin, pimax], i.e.
pi**- pi*<< pimax-pimin (i=1,…,n). The nature of
intervals [pi*, pi**] is defined by methods for
determination of pi (i=1,…,n). Value pi can be
determined by measurements (if possible) or
by calculations for functions fi(m1, m2,…, mk)
(i=1,…,n). In the first case the interval [pi*,pi**]
displays the measurement error, in the second
case this interval displays either calculations
errors or model biases (or, may be, a
combination of these two kinds of errors).
Shift away from “clear”(“accurate”)
values p1, p2,…, pn towards fuzzy values P1,
P2, …, Pn (fuzzification). This fuzzification is
based on the intervals [pimin, pimax] and the
corresponding intervals [pi*, pi**] (i=1,…,n).
Exactly from the point of view of fuzzification
the cases when pi*≤ pimin or pimax ≤ pi** are very
important. The last step of this fuzzification
may be shift away from fuzzy values to
linguistic variables (it may be done for the
convenience of the decisionmaker, but
sometimes it is not necessary to do it). The
essential principle of the described above
fuzzification lies in fuzzifying all input values
into fuzzy membership functions by using
formulae obtained by methods of classical
mathematics (or by using experimental data).
Mathematical equations for mechanical,
physical or chemical processes are considered
only as approximate estimates. These
equations have approximate solutions pi =
fi(m1, m2,…, mk) (i=1,…,n) and form basis for
inequalities. Those inequalities, in their turn,
form the base for constructions of
corresponding fuzzy membership functions
[
Such methodology for mathematical modeling of complicated systems is based on combining of fuzzy logic with classical mathematics.
This methodology is very useful for solving
some problems of the explosion-proof [
The first stage for the development of an information model of every system is its structuring. The architecture of complicated system consists of some components (subsystems) and of the hierarchical relationships between these components. Every subsystem is also complicated system. As a matter of fact, hierarchy is the first feature of a system, since only systems with a hierarchical structure can be in principle investigated.
Every component (subsystem) is assotiated
with an object in terms of OOA. This object must
have attributes [
The selection of the attributes (except identifiers) for such object is reduced to a simple procedure of mnemonic naming of fuzzy variables P1, P2, …, Pn. The methodology for determining these variables is described above. So value of every attribute of the object (again except identifiers) varies from 0 to 1 as the value of a fuzzy variable.
The paper [
Proposed methodology for mathematical and information modeling of complicated systems is very useful for developing of decision support systems (DSS) for automated control system (ACS) when the control object is complicated system.
Thus the developing of DSS is based on
combining of two decision-making models: the
model of choice under uncertainty (based on
fuzzy logic) and the classical model (based on
classical mathematics including classical
numerical methods). Such combining makes it
possible to exclude the participation of evaluators
(experts) in the process of the DSS development.
This is rather important since the difficulties
associated with expert evaluations in organizing
decision-making under uncertainty are well
known [
This methodology is fully adequate for modern technological processes and technical systems. It is used successfully for enlargement and improvement of DSS for explosion-proof of the grain processing enterprises of different types.
Decision support systems for explosionproof of the grain processing enterprises of different types (elevators, flour milling plants, compound feed plants) are enlarged and improved by consistent using of described above items. Original software for real time control of risky situations is created.